Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2023 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(7,16,18,35)(8,20)(9,24)(11,27,28,17)(12,30)(13,21)(14,25)(19,22,26,36)(23,31)(29,32,34,33);(1,2,3,4)(5,6)(7,17,18,27)(8,21)(9,25)(10,15)(11,16,28,35)(12,31)(13,20)(14,24)(19,33,26,32)(22,34,36,29)(23,30);(5,7)(6,11)(8,22)(9,26)(10,18)(12,32)(13,33)(14,34)(15,28)(16,31)(17,30)(19,24)(20,35)(21,27)(23,36)(25,29)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249869, #72249871, #72249882, … |
| maximal normal subgroup of | #72249869, #72249871, #72249882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55318);